Articles of abstract algebra

Question about a projective module in a direct sum

I have a question about my method of proof for proving a simple fact about projective modules. I have a feeling my idea is wrong and I was hoping some one could point out where the mistake is. Let $R$ be a commutative ring with identity. Let $P$ be a projective $R$-module. Prove that there […]

Von Dyck's theorem (group theory)

Did anyone find a proof of this theorem? I can’t find it on the Internet. The theorem is : Let $X$ be a set and let $R$ be a set of reduced words on $X$. Assume that a group $G$ has the presentation $\langle X \mid R \rangle$. If $H$ is any group generated by […]

Confusing partitions of $S_5$ in two different sources

I am trying to understand the partitions of $S_5$ created by it’s conjugacy classes but two sources have two different partitions. Source 1: Source 2: So, for example, in the first table, the partition for cycle structure ()()()()() i.e. $5$ $1$-cycle is $5+0+0+0+0$ but in the second table it is $1+1+1+1+1$. Could anyone please help […]

Where is the “relation” here?

Looking at the definition of a monoid it says that: A monoid is a set that is closed under an associative binary operation and has an identity element $I \in S$ such that for all $a \in S$, $I a = a I =a$ But what does $I a$ mean here? I mean it’s just […]

Showing isomorphism between two rings of fractions

Take $R = \mathbb{Z}$ and fix a set of nonzero integers $X = \{n_{1},n_{2},\dots, n_{k} \}$. Now, let $$D = \{n_{1}^{t_{1}}n_{2}^{t_{2}}\cdots n_{k}^{t_{k}} \mid t_{i}\geq 0 \, \text{for all}\, i,\ t_{i}\neq 0\, \text{for at least one}\, i\}$$ i.e., $D$ is a semisubgroup generated by $X$. I need to show that $D^{-1}\mathbb{Z}$ is isomorphic to another ring […]

Is the cuspidal curve $\mathcal{M}$ is a coarse moduli space for lines in $\mathbb{C}^2$?

As the question suggests, is the cuspidal curve $\mathcal{M}$ a coarse moduli space for lines in $\mathbb{C}^2$? I’m inclined to believe the answer is no, but all attempts at proving it so far have seemed not fruitful…

Normal extensions (a question about the definition)

The definition of a normal extension in the book “Abstract algebra” is : If $K$ is an algebric extension of $F$ which is the splitting field over $F$ for a collection of polynomials $f(x)\in F[x]$ then $K$ is called a normal extension I think that there is something here I don’t understand: If $K$ is […]

Dimension of a splitting field

Given a field $F$ and a polynomial $P \in F[x]$ such that $P$ is irreducible over $F$. Let $L_P$ be the splitting field of $P$ and $F$. Does $\operatorname{dim}_F({L_F}) = \deg(P)$ hold? If looking for a the minimal polynomial of $\alpha$ in a field $F$ is it sufficient to find a polynomial $P$ which is […]

How do I prove the uniqueness of additive identity?

First, suppose $i_1$ and $i_2$ are additive identity in ring R. From the definition of “additive identity” $a+i_1=a$ such that there is $a$ $\in$ $R$, including for $a=i_2$, so $i_2+i_1=i_2$. But $i_2$ is also an additive identity, so there $a \in R$ making $a+i_2=a$. Then $a=i_1$. Then $i_1+i_2=i_1$. Since this is a ring, addition is […]

Additive identity in $\mathbb{Z}_{7}$

The problem consists of two parts: 1)Prove that for some integer $m,$ $3^{m} = 1$ in the integers modulo 7, $\mathbb{Z}_{7}$, which i have done. The second part asks to use this fact to deduce that $7$ divides $1 + 3^{2001}.$ The way i look at it, we look at $1 + 3^{2001}$ in $\mathbb{Z}_{7}$ […]